Is this little toy known ?
Let $E$ be some Banach space, and let $K$ be the closed unit ball of its dual, endowed with the weak-star topology. Also, let $j:E$ $\rightarrow$ $C(K)$ be the natural embedding. Then, if $\pi$ :$E$ $\rightarrow$ $C(K)$ is onto, one must have $\left\Vert \pi-j\right\Vert $ > 1. [Applying this to $E=$ $\ell^{1}$, or to $E=C[0,1]$ (eventually, via Milutin) would be interesting, I think.]
